# Description

Simultaneously performs a non-linear, least squares fit using the Levenberg-Marquardt algorithm of one or more Gaussian singlets, one or more Lorentzian singlets, one or more Gaussian multiplets, and/or a polynomial to one dimensional spectral profiles. A description of the fitting algorithm may be found in AIPS++ Note 224 [1] and in Numerical Recipes [2] . A Gaussian/Lorentzian singlet is a Gaussian/Lorentzian function whose parameters (amplitude, center position, and width) are all independent from any other parameters that may be simultaneously fit. A Gaussian multiplet is a set of two or more Gaussian functions in which at least one (and possibly two or three) of the parameters of each function depend on the parameters of another, single, reference function in the multiplet. For example: one can specify a doublet in which the amplitude of the first spectral line is 0.6 times the amplitude of the zeroth (reference) spectral line, and/or the center of the first line is 20 pixels from the center of the zeroth line, and/or the fwhm of the first line is identical (in pixels) to that of the zeroth line. There is no limit to the number of components one can specify for a multiplet, except of course that the number of parameters to be fit should be significantly less than the number of data points. There can only be a single reference profile in a multiplet, to which the parameters of the other component profiles in the multiplet are tied to.

# AXIS

The axis parameter indicates along which axis the profiles should be fit; a negative value indicates that the spectral axis should be used, or if one does not exist, that the zeroth axis should be used.

# MINIMUM NUMBER OF PIXELS

The minpts parameter indicates the minimum number of unmasked pixels that must be present in order for a fit to be attempted. When multifit=True, positions with too few good points will be masked in any output images.

# ONE FIT OF REGION AVERAGE OR PIXEL BY PIXEL FIT

The multifit parameter indicates if profiles should be fit at each pixel position in the selected region (true), or if the spectral profiles in that region should be averaged together and the fit done to that averaged spectral profile (false).

# POLYNOMIAL FITTING

The order of the polynomial to fit is specified via the poly parameter. If poly<0, no polynomial fit will be attempted. No initial estimates of coefficients can be specified; these are determined automatically.

# Gaussian SINGLET FITTING

The ngauss parameter specifies the maximum number of Gaussian singlets to be fitted, if no estimates are specified by the profile's initial parameter estimates pampest, pcenterest and pfwhmest (the so-called p*est parameters)[a] or by an estimates file, and if gmncomps=0 or is empty. The initial estimates of the parameters for these Gaussians will be determined automatically in this case. If it deems it appropriate, the fitter will fit fewer Gaussians than this number.  If the estimates parameter is not specified, or the p*est parameters are not specified and ngauss=0, gmncomps is empty or 0, and poly<0, an error will occur as this indicates there is nothing to fit.

One can specify initial estimates of Gaussian singlet parameters via an estimates file or the pampest, pcenterest, pfwhmest, and optionally, the pfix parameters. The latter is the recommended way to specify these estimates as support for estimates files may be deprecated in the future. No matter which option is used, an amplitude initial estimate must always be nonzero. A negative fwhm estimate will be silently changed to positve.

## SPECIFYING INITIAL ESTIMATES FOR Gaussian AND Lorentzian SINGLETS (RECOMMENDED METHOD)

One may specify initial estimates via the pampest, pcenterest, and pfwhmest parameters. In the case of a single Gaussian or Lorentzian singlet, these parameters can be scalar numbers. pampest must be specified in image brightness units, pcenterest must be given in the number of pixels from the zeroth pixel, and pfwhmest must be given in pixels. Optionally pfix can be specified and in the case of a single Gaussian or Lorentzian singlet, it can be a string. The string indicates which parameters should be held constant during the fit. Any combination of "p" (amplitude), "c" (center), or "f" (fwhm) is allowed; e.g. pfix="pc" means fix both the amplitude and center of a Gaussian/Lorentzian profile during the fit. In the case of more than one Gaussian and/or Lorentzian singlets, these parameters must be specified as arrays of numbers. The length of the arrays indicates the number of singlets to fit and the array length must be the same for all the p*est parameters.

If no parameters are to be fixed for any of the singlets, pfix can be set to the empty string. However, if at least one parameter of one singlet is to be fixed, pfix must be an array of strings and have a length equal to that of the p*est arrays. Singlets which are not to have any parameters fixed should be represented as an empty string in the pfix array. So, for example, if one desires to fit three singlets and fix the fwhm of the middle one, one must specify pfix=["", "f", ""], the other two empty strings indicating that no parameters of the zeroth and second singlet should be held constant.

In the case of multifit=True, the initial estimates, whether from the p*est parameters or from a file (see below), will be applied to the location of the first fit. This is normally the bottom left corner of the region selected. If the region is masked, or does not contain enough good points to perform a fit, or if the attempted fit fails, the fitting proceeds to the next pixel, with the pixel value of the lowest numbered axis changing the fastest. Once a successful fit has been performed, subsequent fits will use the results of the fit of a nearest pixel, for which a previous fit was successful, as the initial estimates for the parameters at the current location. The fixed parameter string pfix will be honored for every fit performed when multifit=True.

One specifies what type of PCF profile to fit via the pfunc parameter. A PCF function is one that can be parameterized by a peak, center, and FWHM, as both Gaussian and Lorentzian singlets can. If all singlets to be fit are Gaussians, one can set pfunc equal to the empty string and all snglets will be assumed to be Gaussians. If at least one Lorentzian is to be fit, pfunc must be specified as a string (in the case of a single singlet) or an array of strings (in the case of multiple singlets). The position of each string corresponds to the positions of the initial estimates in the p*est and pfix arrays. Minimal match ("g", "G", "l", or "L") is supported. So, if one wanted to simultaneously fit two Gaussian and two Lorentzian singlets, the zeroth and last of which were Lorentzians, one would specify pfunc=["L", "G", "G", "L"].

## ESTIMATES FILE FOR Gaussian SINGLETS (NONRECOMMENDED METHOD)

Initial estimates for Gaussian singlets can be specified in an estimates file via the estimates parameter, which contains the name of the file. Estimates files may be deprecated in the future in favor of the p*est parameters, so it is recommended users use those parameters instead. To use an estimates file, the p*est parameters must be 0 or empty and mgncomps must be 0 or empty. Only Gaussian singlets can be specified in an estimates file. If one desires to fit one or more Gaussian multiplets and/or one or more Lorentzian singlets simultaneously, the p*est parameters must be used to specify the initial parameters of all Gaussian singlets to fit; one cannot use an estimates file in this case. If an estimates file is specified, a polynomial can be fit simultaneously by specifying the poly parameter. The estimates file must contain initial estimates of parameters for all Gaussian singlets to be fit. The number of Gaussian singlets to fit is given by the number of estimate input lines in the file. The file can contain comments which are indicated by a "#" at the beginning of a line. All non-comment lines will be interpreted as initial estimates. The format of such a line is:

[peak intensity], [center], [fwhm], [optional fixed parameter string]

The first three values are required and must be numerical values. The peak intensity must be expressed in image brightness units, while the center must be specified in pixels offset from the zeroth pixel, and fwhm must be specified in pixels. The fourth value is a character string and it is optional. If present, it represents the parameter(s) that should be held constant during the fit. Any combination of the characters 'p' (peak), 'c' (center), and 'f' (fwhm) are permitted, e.g. "fc" means hold the fwhm and the center constant during the fit. Fixed parameters will have no errors associated with them. Here is an example file:

# estimates file indicating that two Gaussians should be fit
# first Gaussian estimate, peak=40, center at pixel number 10.5, fwhm = 5.8 pixels, all parameters allowed to vary during
# fit
40, 10.5, 5.8
# second Gaussian, peak = 4, center at pixel number 90.2, fwhm = 7.2 pixels, hold fwhm constant
4, 90.2, 7.2, f
# end file

# Gaussian MULTIPLET FITTING

Any number of Gaussian multiplets, each containing any number of two or more components, can be simultaneously fit, optionally with a polynomial and/or any number of Gaussian and/or Lorentzian singlets, the only caveat being that the number of parameters to be fit should be significantly less than the number of data points. The gmncomps parameter indicates the number of multiplets to fit and the number of components in each multiplet. In the case of a single multiplet, an integer (>1) can be specified. For example, mgncomps=4 means fit a single quadruplet of Gaussians. In the case of 2 or more multiplets, an array of integers (all >1) must be specified. For example, gmncomps=[2, 4, 3] means 3 separate multiples are to be fit, the zeroth being a doublet, the first being a quadruplet, and the second being a triplet.

Initial estimates of all Gaussians in all multiplets are specified via gmampest, gmcenterest, and gmfwhmest (the so-called gm*est parameters) [b] parameters, which must be arrays of numbers. The input order starts with the zeroth component of the zeroth multiplet to the last component of the zeroth multiplet, then the zeroth component of the first multiplet to the last component of the first multiplet, etc to the zeroth component of the last multiplet to the last element of the last multiplet. The zeroth element of a multiplet is defined as the reference component of that multiplet and has the special significance that it is the profile to which all constraints of all other profiles in that multiplet are referenced (see below). So, in our example of gmncomps=[2, 4, 3], gmampest, gmcenterest, and gmfwhmest must each be nine (the total number of individual Gaussian profiles summed over all multiplets) element arrays. The zeroth, second, and sixth elements represent parameters of the reference profiles in the zeroth, first, and second multiplet, respectively.

The fixed relationships between the non-reference profile(s) and the reference profile of a multiplet are specified via the gmampcon, gmcentercon, and gmfwhmcon parameters. At least one, and any combination, of constraints can be specified for any non-reference component of a multiplet. The amplitude ratio of a non-reference line to that of the reference line is set in gmampcon. The ratio of the fwhm of a non-reference line to that of the reference line is set in gmfwhmcon. The offset in pixels of the center position of a non-reference line to that of the reference line is set in gmcentercon. In the case where a parameter is not constrained for any non-reference line of any multiplet, the value of the associated parameter must be 0. In the case of a single doublet, a constraint may be specified as a number or an array of a single number. For example, mgncomps=2 and gmampcon=0.65 and gmcentercon=[32.4] means there is a single doublet to fit where the amplitude ratio of the first to the zeroth line is constained to be 0.65 and the center of the first line is constrained to be offset by 32.4 pixels from the center of the zeroth line. In cases of a total of three or more Gaussians, the constraints parameters must be specified as arrays with lengths equal to the total number of Gaussians summed over all multiplets minus the number of reference lines (one per multiplet, or just number of multiplets, since reference lines cannot be constrained by themselves). In the cases where an array must be specified but a component in that array does not have that constraint, 0 should be specified. Here's an example:

gmncomps=[2, 4, 3]
gmampcon= [ 0 , 0.2, 0 , 0.1, 4.5, 0 ]
gcentercon=[24.2, 45.6, 92.7, 0 , -22.8, -33.5]
gfwhmcon=""

In this case we have our previous example of one doublet, one quadruplet, and one triplet. The first component of the doublet has the constraint that its center is offset by 24.2 pixels from the zeroth (reference) component. The first component of the quadruplet is constrained to have an amplitude of 0.2 times that of the quadruplet's zeroth component and its center is constrained to be offset by 45.6 pixels from the reference component. The second component of the quadruplet is constained to have its center offset by 92.7 pixels from the associated reference component and the third component is constrained to have an amplitude of 0.1 times that of the associated reference component. The first component of the triplet is constrained to have an amplitude of 4.5 times that of its associated reference component and its center is constrained to be offset by -22.8 pixels from the reference component's center. The second component of the triplet is constrained to have its center offset by -33.5 pixels from the center of the reference component. No lines have FWHM constraints, so the empty string can be given for that parameter. Note that using 0 to indicate no constraint for line center means that one cannot specify a line centered at the same position as the reference component but having a different FWHM from the reference component. If you must specify this very unusual case, try using a very small positive (or even negative) value for the center constraint.

Note that when a parameter for a line is constrained, the corresponding value for that component in the corresponding gm*est array is ignored and the value of the constrained parameter is automatically used instead. So let's say, for our example above, we had specified the following estimates:

gmampest = [ 1, .2, 2, .1, .1, .5, 3, 2, 5]
gmcenterest = [20, 10 , 30, 45.2, 609 , -233, 30, -859, 1]

Before any fitting is done, the constraints would be taken into account and these arrays would be implicitly rewritten as:

gmampest = [ 1, .2, 2, .4, .1, .2, 3, 13.5, 5 ]
gmcenterest = [20, 44.2, 30, 75.6, 127.7, -233, 30, 7.2, -3.5]

The value of gmfwhmest would be unchanged since there are no FWHM constraints in this example.

In addition to be constrained by values of the reference component, parameters of individual components can be fixed. Fixed parameters are specified via the gmfix parameter. If no parameters are to be fixed, gmfix can be specified as the empty string or a zero element array. In the case where any parameter is to be fixed, gmfix must be specified as an array of strings with length equal to the total number of components summed over all multiplets. These strings encode which parameters to be fixed for the corresponding components. If a component is to have no parameters fixed, an empty string is used. In other cases one or more of any combination of parameters can be fixed using "p", "c", and/or "f" described above for fixing singlet parameters. There are a couple of special cases to be aware of. In the case where a non-reference component parameter is constrained and the corresponding reference component parameter is set as fixed, that parameter in the non-reference parameter will automatically be fixed even if it was specified not to be fixed in the gmfix array. This is the only way the constraint can be honored after all. In the converse case of when a constrained parameter of a non-reference component is specified as fixed, but the corresponding parameter in the reference component is not specified to be fixed, an error will occur. Fixing an unconstrained parameter in a non-reference component is always legal, as is fixing any combination of parameters in a reference component (with the above caveat that corresponding constrained parameters in non-reference components will be silently held fixed as well).

The same rules that apply to singlets when multifit=True apply to multiplets.

# LIMITING RANGES FOR SOLUTION PARAMETERS

In cases of low (or no) signal to noise spectra, it is still possible for the fit to converge, but often to a nonsensical solution. The astronomer can use her knowledge of the source to filter out obviously spurious solutions. Any solution which contains a NaN value as a value or error in any one of its parameters is automatically marked as invalid.

One can also limit the ranges of solution parameters to known "good" values via the goodamprange, goodcenterrange, and goodfwhmrange parameters. Any combination can be specified and the limit constraints will be ANDed together. The ranges apply to all PCF components that might be fit; choosing ranges on a component by component basis is not supported. If specified, an array of exactly two numerical values must be given to indicate the range of acceptable solution values for that parameter. goodamprange is expressed in terms of image brightness units. goodcenterrange is expressed in terms of pixels from the zeroth pixel in the specified region. goodfwhmrange is expressed in terms of pixels (only non-negative values should be given for FWHM range endpoints). In the case of a multiple-PCF fit, if any of the corresponding solutions are outside the specified ranges, the entire solution is considered to be invalid.

In addition, solutions for which the absolute value of the ratio of the amplitude error to the amplitude exceeds 100 or the ratio of the FWHM error to the FWHM exceeds 100 are automatically marked as invalid.

# INCLUDING STANDARD DEVIATIONS OF PIXEL VALUES

If the standard deviations of the pixel values in the input image are known and they vary in the image (e.g. they are higher for pixels near the edge of the band), they can be included in the sigma parameter. This parameter takes either an array or an image name. The array or image must have one of three shapes:

1. the shape of the input image,
2. the same dimensions as the input image with the lengths of all axes being one except for the fit axis which must have length corresponding to its length in the input image, or
3. be one dimensional with length equal the the length of the fit axis in the input image.

In cases 2 and 3, the array or pixels in sigma will be replicated such that the image that is ultimately used is the same shape as the input image. The values of sigma must be non-negative. It is only the relative values that are important. A value of 0 means that pixel should not be used in the fit. Other than that, if pixel A has a higher standard deviation than pixel B, then pixel A is noisier than pixel B and will receive a lower weight when the fit is done. The weight of a pixel is the usual:

weight = $\frac{1}{\sigma^2}$

In the case of multifit=F, the sigma values at each pixel along the fit axis in the hyperplane perpendicular to the fit axis which includes that pixel are averaged and the resultant averaged standard deviation spectrum is the one used in the fit. Internally, sigma values are normalized such that the maximum value is 1. This mitigates a known overflow issue.

One can write the normalized standard deviation image used in the fit but specifying its name in outsigma. This image can then be used as sigma for subsequent runs.

# RETURNED DICTIONARY STRUCTURE

The dictionary returned (if wantreturn=True) has a (necessarily) complex structure. First, there are keys "xUnit" and "yUnit" whose values are the abscissa unit and the ordinate unit described by simple strings. Next there are arrays giving a broad overview of the fit quality. These arrays have the shape of the specified region collapsed along the fit axis with the axis corresponding to the fit axis having length of 1:

• ATTEMPTED: a boolean array indicating which fits were attempted (e.g. if too few unmasked points, a fit will not be attempted)
• CONVERGED: a boolean array indicating which fits converged. False if the fit was not attempted
• VALID: a boolean array indicating which solutions fall within the specified valid ranges of parameter space (see section LIMITING RANGES FOR SOLUTION PARAMETERS for details)
• NITER: an int array indicating the number of iterations for each profile, a negative value indicates the fit did not converge
• NCOMPS: the number of components (Gaussian singlets + Lorentzian singlets + Gaussian multiplets + polynomial) fit for the profile, a negative value indicates the fit did not converge
• DIRECTION: a string array containing the world direction coordinate for each profile

There is a "type" array having number of dimensions equal to the number of dimensions in the above arrays plus one. The shape of the first n-1 dimensions is the same as the shape of the above arrays. The length of the last dimension is equal to the number of components fit. The values of this array are strings describing the components that were fit at each position ("POLYNOMIAL", "Gaussian" in the case of Gaussian singlets, "Lorentzian" in the case of Lorentzian singlets, and ""Gaussian MULTPLET").

If any Gaussian singlets were fit, there will be a subdictionary accessible via the "gs" key which will have subkeys "amp", "ampErr", "center", "centerErr", "fwhm", "fwhmErr, "integral", and "integralErr". Each of these arrays will have one more dimension than the overview arrays described above. The shape of the first n-1 dimensions will be the same as the shape of the arrays described above, while the final dimension will have length equal to the maximum number of Gaussian singlets that were fit. Along this axis will be the corresponding fit result or associated error (depending on the array's associated key) of the fit for that singlet component number. In cases where the fit did not converge, or that particular component was excluded from the fit, a value of NAN will be present.

If any Lorentzian singlets were fit, their solutions will be accessible via the "ls" key. These arrays follow the same rules as the "gs" arrays described above.

If any Gaussian multiplets were fit, there will be subdictionaries accessible by keys "gm0", "gm1", ..., "gm{n-1}" where n is the number of Gaussian muliplets that were fit. Each of these dictionaries will have the same arrays described above for Gaussian singlets. The last dimension will have length equal to the number of components in that particular multiplet. Each pixel along the last axis will be the parameter solution value or error for that component number in the multiplet, e.g. the zeroth pixel along that axis contains the parameter solution or error for the reference component of the multiplet.

The polynomial coefficient solutions and errors are not returned, although they are logged.

# OUTPUT IMAGES

In addition to the returned dictionary, optionally one or more of any combination of output images can be written. The model and residual parameters indicate the names of the model and residual images to be written; blank values inidcate that these images should not be written.

One can also write none, any or all of the solution and error images for Gaussian singlet, Lorentzian singlet, and Gaussian multiplet fits via the parameters amp, amperr, center, centererr, fwhm, fwhmerr, integral, and integralerr when doing multi-pixel fits. These images simply contain the arrays described for the associated parameter solutions or errors described in previous sections. In the case of Lorentzian singlets, "_ls" is appended to the image names, in the case of Gaussian multiplets, "_gm" is appended. Pixels for which fits were not attempted or did not converge will be masked as bad. The last axis of these images is a linear axis and repesents component number (and is named accordingly). In the case where multiple Gaussian singlets and/or Lorentzians are fitted, the image names are further appended with an underscore and the relevant component number ("_0", "_1", etc). In the case of Gaussian multiplets, the image names are appended with an underscore, followed by the number of the relevant multiplet group, followed by an underscore, followed by the number of the component in that group (e.g., "image_gm_3_4" represents component number 4 of multiplet group number 3). Pixels for which fits were not attempted, did not converge, or converged but have values of NaN (not a number) or INF (infinity) will be masked as bad.

Writing analogous images for polynomial coefficients is not supported.

Citation Number 1 [Brouw, Wim, 1999 (Web)]
Citation Number 2 W.H. Press et al 1988., Cambridge University Press
Footnote Number a p*est refers to the various task parameters pampest, pcenterest, and pfwhmest

Footnote Number b gm*est refers to the various task parameters gmampest, gmcenterest, and gmfwhmest